Fundamentally, there are two types of Verge wallets you can use; QT and/or Electrum. For everyday users, we recommend you download an Electrum wallet, e.g. MyVergies.
Electrum is an open-source Bitcoin lite wallet solution. Verge has forked the Electrum wallet codebase and updated to have it work with the Verge blockchain.
A Verge Electrum wallet is a wallet that is focused on speed and low resource usage. Electrum wallets use a technology called, Simple Payment Verification (SPV), that verifies transactions on the Verge blockchain without needing to download the entire blockchain. The SPV technology allows average transaction confirmation times to drop to ~5 seconds (instead of ~30 seconds for non-Electrum). For more general questions about Electrum, please go to their FAQ page.
QT is a cross-platform application and UI framework for developers using the C++ programming language. It allows developers to write code in C++ and then release their software to various software and hardware platforms. Per Wikipedia, the toolkit was named QT because "the letter Q looked appealing in Haavard's Emacs typeface, and 't' was inspired by Xt, the X toolkit".
QT wallets require that you download the entire Verge blockchain to your computer. In other words, your computer must download a record of every single Verge transaction that has ever occurred. Currently, the Verge blockchain is ~1GB and will continue to climb as more users make transactions. If you are using a QT wallet, make sure your device has enough extra storage to handle it.
Every time you open your QT wallet, the wallet will synchronize and download the latest blockchain data to your computer. If you decide to use a QT wallet, you will most likely want to use it on a regular basis; or it could take an hour or more (depending on how long it has been since you opened your wallet last) to download the latest blockchain updates.